bree-AWN
/bɹiɑn/
Here is a clip of characters from the anime talking about Brionne. Note that they place the stress on the second syllable, so it's more similar to "beyond" (bee-YOND) than "neon" (NEE-awn), at least under a standard American accent.
I know that the anime sometimes makes mistakes, but this pronunciation is pretty regular based on the spelling. According to the DB and Bulbapedia, Brionne's English name may have originated from the French sissonne, a ballet step, which fits Brionne's theme of being a dancer. The -onne suffix typically becomes an "awn" sound when anglicized; compare with the female name Yvonne.